Author: UrsFormat: MarkdownItexhave created _[[extension of distributions]]_ with the statement of the characterization of the space of _point-extensions_ of distributions of finite degree of divergence: [here](https://ncatlab.org/nlab/show/extension+of+distributions#SolutionSpaceOfPointExtensions)
This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of "[[causal perturbation theory]]". Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to [Epstein-Glaser 73](https://ncatlab.org/nlab/show/The+Role+of+locality+in+perturbation+theory). But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.

have created extension of distributions with the statement of the characterization of the space of point-extensions of distributions of finite degree of divergence: here

This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of “causal perturbation theory”. Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to Epstein-Glaser 73. But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.

Author: zskodaFormat: MarkdownItexI do not get it. What is the definition ? The idea section is an ill-defined requirement, I do not see how it gives a definition. The two of the 3 references given do not give a general definition but some proposal for a realization of an extension in some situation.
I gave an ad hoc definition (and added some lines in the idea section) for the special case when $X\hookrightarrow\hat{X}$ is an open embedding, using the standard definition of the restriction of distributions in that case. I also added the redirect [[restriction of distributions]] and
[[restriction of a distribution]]. According to the general conventions of the $n$Lab this entry should be called [[extension of a distribution]] rather than plural in second half [[extension of distributions]]. This is what I added:
>Regarding that we the distributions are not the maps from the underlying space we need to replace pre-composition $X\to\tilde{X}$ by the appropriate pullback operation; if $X\to\tilde{X}$ is an open embedding this will be the operation of restriction of distributions dual to the operation of extension by zero of test functions.
> For an inclusion of two open sets on a manifold $U\subset V$ there is an operator of _extension by zero_ $E_{V U}: C^\infty_0(U)\to C^\infty_0(V)$ where $E_{V U}(f)(x) = f(x)$ if $x\in U$ and $E_{U V}(f)(x) = 0$ otherwise.
>The __restriction__ $\rho_{U V} : \mathcal{D}'(V)\to \mathcal{D}'(U)$ is then defined by
> $$
\langle \rho_{U V}(\phi), f\rangle = \langle \phi, E_{V U} f\rangle,
\,\,\,\,\,\,\,f\in C^\infty_0(U),\,\phi\in \mathcal{D}'(V).
$$
> Now the diagram in the idea section makes sense in the following way: for an open embedding $X\hookrightarrow\hat{X}$, $\hat{u}\in\mathcal{D}'(\hat{X})$ __extends__ $u\in \mathcal{D}'(X)$ if $\rho_{X \hat{X}}(\hat{u}) = u$.

I do not get it. What is the definition ? The idea section is an ill-defined requirement, I do not see how it gives a definition. The two of the 3 references given do not give a general definition but some proposal for a realization of an extension in some situation.

Regarding that we the distributions are not the maps from the underlying space we need to replace pre-composition X&rightarrow;X˜X\to\tilde{X} by the appropriate pullback operation; if X&rightarrow;X˜X\to\tilde{X} is an open embedding this will be the operation of restriction of distributions dual to the operation of extension by zero of test functions.

For an inclusion of two open sets on a manifold U&subset;VU\subset V there is an operator of extension by zeroEVU:C0∞(U)&rightarrow;C0∞(V)E_{V U}: C^\infty_0(U)\to C^\infty_0(V) where EVU(f)(x)=f(x)E_{V U}(f)(x) = f(x) if x&Element;Ux\in U and EUV(f)(x)=0E_{U V}(f)(x) = 0 otherwise.
The restrictionρUV:&Dscr;′(V)&rightarrow;&Dscr;′(U)\rho_{U V} : \mathcal{D}'(V)\to \mathcal{D}'(U) is then defined by

Now the diagram in the idea section makes sense in the following way: for an open embedding X&hookrightarrow;X^X\hookrightarrow\hat{X}, u^&Element;&Dscr;′(X^)\hat{u}\in\mathcal{D}'(\hat{X})extendsu&Element;&Dscr;′(X)u\in \mathcal{D}'(X) if ρXX^(u^)=u\rho_{X \hat{X}}(\hat{u}) = u.

Author: UrsFormat: MarkdownItexAs you say, what you made explicit is the standard definition of restriction of distributions, a special case of the pullback of distributions along a submersion [here](https://ncatlab.org/nlab/show/pullback+of+a+distribution#PullbackOfDistributionAlongSubmersion).
Regarding the naming convention:
* "extension of distributions" is singular, it refers to a single function $\widehat{(-)} :\mathcal{D}'(X) \to \mathcal{D}'(Y)$
* "extension of a distribution" is also singular, but it refers a single $u \mapsto \widehat u$.
I think the former pattern should be preferred, though sometimes I do use the second.
The plural would be "extensions of distributions" and "extensions of a distribution", respectively.

As you say, what you made explicit is the standard definition of restriction of distributions, a special case of the pullback of distributions along a submersion here.

Regarding the naming convention:

“extension of distributions” is singular, it refers to a single function (−)^:&Dscr;′(X)&rightarrow;&Dscr;′(Y)\widehat{(-)} :\mathcal{D}'(X) \to \mathcal{D}'(Y)

“extension of a distribution” is also singular, but it refers a single u&map;u^u \mapsto \widehat u.

I think the former pattern should be preferred, though sometimes I do use the second.

The plural would be “extensions of distributions” and “extensions of a distribution”, respectively.